To obtain the bridge function, we need to solve the functional equation (REF ). However, directly solving (REF ) in a rich function space can be ill-posed (see discussion in [1]}). To address this, recent works [2]}, [3]} minimize the following regularized loss \(\mathcal {L}_{\mathrm {PV}}\)  to obtain an estimate of the bridge function \(\hat{h}\) :

h = hHh LPV(h),    LPV(h) = [Y,A,Z](Y - [W|A,Z]h(A, W))2 + (h), 

where \(\mathcal {H}_h\)  is an arbitrary space of continuous functions and \(\Omega (h)\)  is a regularizer on \(h\) . Note that this loss can be interpreted as the deviation of the r.h.s and the l.h.s in (REF ) measured in \(L^2\) -norm with respect to the distribution \({Y,A,Z}\) .
